Question

If $\sec \theta + \tan \theta = p$, obtain the value of $\sec \theta ,\tan \theta ,\sin \theta $ in terms of p.

Answer 1

Hint: Here we use the trigonometry identities and formulae to obtain the values.

“Complete step-by-step answer:”
As we know ${\sec ^2}\theta - {\tan ^2}\theta = 1$then
$
  (\sec \theta + \tan \theta )(\sec \theta - \tan \theta ) = 1 \\
  (\sec \theta - \tan \theta ) = \dfrac{1}{{(\sec \theta + \tan \theta )}} \\
  (\sec \theta - \tan \theta ) = \dfrac{1}{p} \to (1){\text{ [}}\because (\sec \theta - \tan \theta ) = p] \\
  (\sec \theta + \tan \theta ) = p \to (2) \\
 $
Now add equation (1) and (2) we get
$\sec \theta = \dfrac{1}{2}\left( {\dfrac{{{p^2} + 1}}{p}} \right)$
Now subtract equation (1) and (2) we get
$\tan \theta = \dfrac{1}{2}\left( {\dfrac{{{p^2} - 1}}{p}} \right)$
From the values of $\sec \theta $and$\tan \theta $, we know $\dfrac{{\tan \theta }}{{\sec \theta }} = \sin \theta $
Therefore, $\sin \theta = \dfrac{{{p^2} - 1}}{{{p^2} + 1}}$.
Hence, we get the answer.

Note: Whenever such type of questions are always try to start the question with use of the trigonometric identities and use some algebraic formula to find the answer as we know $({a^2} - {b^2}) = (a + b)(a - b)$ that we apply in the question to solve it.